Let F be a field and K sub-field of F.

1. Let A be a matrix that her element in K. Prove that rank(A) over K equals to rank(A) over F.

2. Suppose we have a system of equations so that all the cofetens of the expand matrix of the system are in K. Show that there is a solution over K if and only if there is solution over F. In case that there is solution show that the dimension of affine space(of the solutions) not changing if we think the system over K or over F.




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